Measuring an Almost Ideal Demand System with Generalized Flexible Least Squares

Michael C. Poray

Kenneth A. Foster

and

Jeffery H. Dorfman

Abstract

Structural change in meat consumption has been the focus of many researchers during the lasttwo decades. In this paper we develop a dynamic linear Almost Ideal Demand System (AIDS)model from a cost function that allows for time varying parameters. This model is consistentwith inertia in the parameters of the cost and indirect utility functions. It allows for persistentpreferences which may arise from cultural biases, lifestyles, peer pressure, etc. An empiricalapplication is conducted with US meat consumption and price data using a generalized system offlexible least squares, Generalized Flexible Least Squares (GFLS). GFLS allows parameters toevolve slowly over time through incorporating of penalties in fluctuations. Estimated quarterlyelasticities were subjected to additional analysis to determine how highly they were related to theBrown and Schrader Cholesterol Index and relative prices. The combined results support thatthe movements of elasticities over time are related to both.

Selected paper presented at the American Agricultural Economics Association Annual Meetingin Tampa, Florida July30-August 2, 2000.

Michael C. Poray is a Graduate Research Assistant in the Department of AgriculturalEconomics at Purdue University.

Kenneth A. Foster is a Professor in the Department of Agricultural Economics at PurdueUniversity.

Jeffery H. Dorfman is a Professor in the Department of Agricultural Economics at the Universityof Georgia.

Copyright 2000 by Michael C. Poray, Kenneth A. Foster, and Jeffery H. Dorfman, all rightsreserved. Reader may make verbatim copies of this document for non-commercial purposes byany means, provided this copyright appears on all such copies.

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Measuring and Almost Ideal Demand System with Generalized Flexible Least Squares

Per capita beef consumption in the United States today is nearly the same as it was in the

mid-1960's. The interim period, however, saw a 25 percent increase between 1967 and 1976

followed by a 23 percent decrease from 1976 to 1988. Attempts to explain these wide swings in

meat consumption patterns over the past 25 years have used a variety of techniques and

functional forms. The debate that has evolved is much like the popular lite beer commercial

where one group chants "tastes great" and the other responds "less filling". The two cheers heard

in the meat demand arena are "relative prices" and "changed preferences". Of course the purpose

of the beer commercial is to persuade consumers that the product is less filling and tastes great.

Likewise, this paper attempts to present an analysis that is capable of encompassing an important

role for both relative prices and changing consumer tastes in the context of meat demand.

With the exception of Chavas, the previous structural change models have not accounted

for the persistence of the parameters of preferences over time. That is, any structural change due

to changes in preferences should be expected to occur slowly or smoothly as suggested by

Moschini1. Slowly evolving preferences are consistent with the effects of cultural biases,

lifestyles, peer pressure, and habit formation, all of which prohibit rapid adjustments in utility

function parameters. The switching regression models estimated by Braschler and Moschini and

Meilke (1989) generated fairly discrete changes in parameter values. Both papers identified

significant structural change in the mid-1970's. The results do not provide a means to determine

the relative importance of preference changes and relative prices. The Moschini and Meilke

(1989) paper estimated a statistically significant linear trend in the parameters of an almost ideal

demand system for the four quarters between 1975-IV to 1976-III. For the beef industry, and by

association other livestock, this was a fairly aberrant year. It follows closely on the heels of

3

price controls that had a profound impact on farm level cattle prices and the supply of cattle.

Many farmers held cattle back from slaughter hoping to capitalize on anticipated higher prices

when the price controls were lifted. The effect was an over-supply of cattle, a drastic drop in

cattle prices, and the beginning of a sustained liquidation of the aggregate cattle herd. One

aspect of the available aggregate data is that beef is a perishable commodity and that prices often

adjust to clear whatever quantity is produced. Consequently, it is possible that the mid-70's

represent a phenomena best handled by a dummy variable rather than a permanent change in

structure.

Using different functional forms Moschini and Meilke (1984) and Wohlgenant failed to

find significant structural change in meat demand. Wohlgenant concluded that previous

evidence of structural change was confounded with the specification of functional form.

Chalfant and Alston used a nonparametric test to determine that meat demand data were

consistent with unchanged tastes over time. In another paper, Alston and Chalfant show that

Canadian meat demand data pass the nonparametric test, however, the same data revealed

significant structural change within the context of two parametric forms.

Time-Varying Consumer Preferences

As mentioned above many authors have generated plausible models with either dynamic

coefficients or lagged variables (see Blanciforti and Green, and Eales and Unnevehr for the

latter). In order to introduce time varying coefficients into a demand system some theoretical, as

well as heuristic arguments are valuable. To begin with, consider a world with intertemporally

additive preferences represented by the PIGLOG cost function.

( ) ( ) ( ) ( )pbupaupuc lnln1,ln +−= (1)

1Moschini used a time trend and the Brown and Schrader cholesterol index to mimic smooth preference changes.

4

where a(p) and b(p) are both homogeneous of degree one in prices, p, and u is utility ranging

from zero to one. This is familiar as the cost function which Deaton and Muellbauer used to

develop the almost ideal demand system2. Given Deaton and Muellbauer's parameterizations

and the cost function in (1) it is possible to solve for the indirect utility function

( )( )∏

=

kk

kp

paxu ββ0

ln(2)

where x is total expenditure and is equal to c(u,p) for the utility maximizer.

Typically, structural change is considered to involve changes in parameter values within

a single specification. In the context of the almost ideal demand system this implies that the

parameters of ln a(p) and ln b(p) change over time. Consequently, alternative specifications of

ln a(p) and ln b(p) which are capable of allowing a wide scope of parameter mobility are needed.

Chavas has indirectly implied a form for these new specifications in his earlier Kalman filter

paper. Consider the following

( ) ktjtk j

jtkktk

ktt ppppa lnln2

1lnln 0 ∑∑∑ ++= γαα (3)

with

( ) ttt etF +=+ φφ 1

Where φ is a vector of parameters {α0, …, αk,…, γ11,…, γkk}. The F(t) matrix is the dynamic

equation matrix, and et is a random forcing term. Further Chavas (1983) specifies

( ) ( ) ∏+=k

ktktppapb ββ 0lnln (4)

2 The choice of ln a(p) and ln b(p) determine the characteristic of the demand equations, as well as, the flexibilityof the cost function. Deaton and Muellbauer chose the following forms

( ) kjk k

jkkk

k ppppa lnln2

1lnln 0 ∑∑∑ ++= γαα

5

with

( ) kutG tktktk ∀+=+ ,,1, ββ

Together these specifications allow the own and cross price elasticities to vary over time

with preferences, as well as, price and quantity levels. The result is a linkage of consumption

patterns over time. Thus, the intertemporal separability assumption is somewhat weakened since

estimates of φt+1 and βt+1 will, in general, be calculated as functions of past quantities and prices.

To further develop the model we use equations (1) - (4) and Shepard's Lemma to derive a

system of share equations which resemble the usual static almost ideal demand system, but have

the innovation that the parameters move according to the time paths described for them above.

The ith share equation is given by

++= ∑

t

titjt

jijtitit P

xPw lnln βγα (5)

where Pt is the Stone's price index commonly used to linearize the almost ideal demand system,

and xt is total expenditures on meat. The parameters of this demand system evolve from their

predecessors over time taking into account the inertia in human behavior, and because of the

random forcing terms, can represent any form of structural change if it occurs. Undoubtedly,

constant changes in preferences are occurring, but because the observed data is highly

aggregated it is difficult to model, with standard techniques, anything except very large shifts or

an accumulation of small changes in the same direction. This demand system assumes that

smaller adjustments can be accurately modeled with existing data. The analysis with switching

regimes where either a single shift or constant linear trends were found suggests simplistic

regimes rather than a perpetually evolving system of preferences. With the above analysis, the

( )( ) ( ) ∏+=k

kkPPaPb ββ 0lnln

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parameters could move according to any of a variety of dynamic patterns including, logistical,

random, or linear trend.

Few supporters of a preference change in meat consumption would disagree that such

changes occurred over a long period of time between the 1970's and present. In fact, the

adjustments may not be complete. As prices, nutritional information, and product characteristics

evolve in a dynamic economy decisions made in the past continue to influence the supply and

demand for meat and other products.

In previous meat demand studies significant parameter changes have been discovered.

However, Wohlgenant demonstrated that such statistical results can be dependent on the

specification of the demand system used. Therefore, it is insufficient to simply suggest a time-

varying parameter model and estimate it. Time-varying elasticities can be calculated from the

above almost ideal demand system and these elasticities could vary for two main reasons. First,

due to relative prices, and second, due to preference changes. If the structural changes implied

by the parameter estimates are the result of an incorrect specification, then they should not vary

systematically with any measure of consumer tastes. That is, the elasticities should not be

significantly affected by such a measure, but should vary significantly only with relative price

changes. A straightforward test of the importance of these factors can be implemented by

regressing the elasticities against relative prices and a measure of consumer tastes.

Data

The data used in this analysis are identical to those used by Moschini and Meilke (1989).

They consist of quarterly per capita disappearances and retail prices of beef, chicken, fish and

seafood, and pork. Quantities and prices for chicken are those published by USDA in Poultry

and Egg Situation and Livestock and Poultry Situation. The quantity reflects per capita

7

disappearance of total chicken (young and mature), and the price is that of fryers. Choice beef

and pork quantities and prices came from USDA's Livestock and Meat Situation. Fish and

seafood consumption data are derived from USDC data on personal expenditures and the CPI for

fish and seafood. The sample begins in 1966 and extends through 1987 spanning the mid-1970's,

the period of structural change discovered by previous research. The Brown and Schrader

cholesterol index was chosen to represent potential change in consumer tastes. This index is the

sum of the number of medical journal articles supporting a link between cholesterol intake and

heart disease minus those that do not support such a link. The articles were identified using the

Medline database. Brown and Schrader suggest using the two-quarter lag of this sum in order to

capture the lag time between publication of the article and the impact (via physicians) on

consumers. While this represents a fairly ad hoc distributed lag effect it was deemed satisfactory

for this study since the purpose is to demonstrate a significant link to changing meat

consumption not accurately quantify the marginal impact of another medical journal article.

Empirical Implementation

Adding a disturbance term to the share equations in (5) yields an empirical model of the

type which West and Harrison refer to as a Dynamic Linear Model (DLM). In general terms the

DLM is written

tttt uxy += β (6a)

tttt eF +=+ ββ 1 (6a)

Equation (6a) is commonly called the observation equation and (6b) is referred to as the system

evolution equation.

There are a number of related methods that can be used to estimate the unknown

parameters, βt. In fact, the system estimated by Chavas is similar although his specification is

8

slightly different from the almost ideal demand system posited in the previous section. He used

a standard Kalman filter to estimate time updated parameter values and found structural change

in beef and chicken consumption in the 1970's. Bayesian techniques could also be used if

workable priors can be identified.

The approach chosen in this paper is in a sense a hybrid of the Kalman filter and

Bayesian estimation. The Generalized Flexible Least Squares (GFLS) approach is a multiple

equation extension of the Flexible Least Squares (FLS) technique developed by Kalaba and

Tesfatsion. In fact, Dorfman and Foster used the single equation FLS model to estimate

technical change in a production economics setting. GFLS represents a special case of the

Kalman filter in which updates in parameter estimates are penalized for deviations from previous

values. This slows and smooths parameter movement over time and is consistent with the notion

that preferences tend to be sticky3.

The GFLS algorithm seeks to minimize a loss function which includes both the usual sum

of squared errors from the observation equation (SSE), and the sum of squared dynamic errors

from the evolution equation (SSD). For the GFLS case the loss function is

( ) [ ] [ ] [ ] [ ]∑∑=

−

=++ −′−+−′−=

T

tttttttt

T

tttttttt xyMxyFDFTC

1

1

111,; ββββββµµβ (7)

GFLS solves for the trajectory of β that minimizes the loss function over time. The

matrix Mt is analogous to the variance-covariance matrix in standard generalized least squares

regression. Under the assumption of contemporaneously correlated equation disturbances the ij th

element of M is ( )knuu ji −′ . The parameter µ defines the relative importance of the two terms

in the objective functions. When values of µ approach infinity the GFLS estimates converge to

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constant parameter seemingly unrelated regressions estimates, because µ represents the penalty

for parameter movements over time. Setting µ equal to zero leads to a random coefficients

model analogous to that developed by Singh and Ullah, because there are no penalties for

parameter movement4. As a consequence, there is a trade-off between SSE and SSD which can

be depicted graphically as in Dorfman and Foster's efficiency frontier when the value of µ is

varied from 0 to ∞ . In the classical econometric context µ is the inverse of the prior precision

for a stochastic restriction. Thus, in past applications the specification of µ has been arbitrary.

In order to find a less ad hoc value of µ one could begin with the prior belief of a strict

classical econometrician that both sums of squares will be zero. That is, the parameters are not

time-varying, and the model is the correct specification with zero measurement error and no

omitted variables. While few econometricians actually hold this naive view of the world, it

remains true that this is the prior belief underlying the majority of applied econometric analyses.

The point on the efficiency frontier closest to the origin yields the estimate of µ most consistent

with this naive prior belief. Thus, we propose to choose µ by minimizing the vector norm, in the

efficiency frontier space, that is defined as

2

2

2

2

RC

fls

ols

fls

SSE

SSD

SSE

SSEJ += (8)

where 2olsSSE is the sum of squared errors when the model is estimated with constant parameters

(i.e. SUR), and 2rcSSE is the sum of squared errors when the model is estimated as a seemingly

unrelated random coefficients model, and the fls subscripts imply the counterparts when the

3The reference to slow and smooth parameter dynamics should not mislead one to believe that GFLS is incapable ofallowing for rapid structural shifts. Kalaba and Tesfatsion and Tesfatsion and Veitch have demonstrated the abilityof FLS estimates to track a wide variety of "smooth" and "nonsmooth" (rapid) parameter transitions.4 Specifying the matrix D as other than an identity matrix will result in separate penalties for movement of theelements in βt over time.

10

model is estimated by generalized flexible least squares. The scaling is necessary to prevent the

relative size of the dynamic and measurement errors from skewing the efficiency frontier. The

value of µ chosen by this method represents an interpretable specification in the context of the

constant parameter model in two ways. First, the estimates can be compared to the constant

parameter estimates. Second, it is the model that the data suggests is most consistent with both

constant parameters and a zero SSE.

We maintained the assumptions of weak separability between meat and other

commodities and intertemporal separability to simplify the estimation and remain consistent with

previous meat demand studies. Furthermore, the dynamic equation matrix was assumed to be an

identity. The weighting matrix D in equation (7) was held constant over time. The off-diagonal

elements of this matrix were set to zero and the diagonal elements associated with independent

variables other than quarterly dummies were set to unity. The diagonal elements associated with

the quarterly dummy variables were set to ten million in order to prevent any movement in their

parameter estimates. Quarterly dummy variables with time-varying coefficients could

potentially pick up all of the variation in the shares and lead to unsatisfactory estimates of the

remaining coefficients.

To implement this estimation a search routine was used to find the value of µ which

minimized the norm defined in equation (8). The routine parametrically varied µ from zero to

19,0355 to minimize J. The value of µ that minimized J was found to be 2.14. This is appears to

be closer to the random coefficients model than the SUR model and is close in magnitude to the

value used by Dorfman and Foster. Figures 1 and 2 show the movement of the own price

coefficients estimated by the GFLS/AIDS approach. Notice that both the beef and chicken

5 19,035 was the largest value of µ that produced non-zero estimates for the coefficients.

11

coefficients follow upward trends, with the own price coefficient for chicken increases more than

four times from 1974-I to 1975-I. The own price coefficient for pork falls until 1980-81 and then

increases over the remaining periods in the sample to higher than initial levels. The own price

coefficient for fish trends down over the entire sample with some large changes between 1974

and 1976. Table 1 presents the mean, minimum, maximum, and standard deviations of all

parameter estimates. The model fit was very satisfactory. For all equations in the system, the

squared correlation between actual and predicted expenditure shares exceeded 0.98.

Another interesting artifact of the chosen estimation method is the opportunity to evaluate

the bias in structural change over time. Antle and Capablo present a good discussion of the

multifactor dual (cost function) measure of bias first derived by Binswanger (1974, 1978).

Furthermore, Antle and Capalbo show that Binswanger's bias measure contains a confounding

scale effect when technology (preference) is not homothetic. They suggest the correct scale

adjustment that leaves a pure measure of bias. An analogous approach works here since the

GFLS/AIDS model is derived from the expenditure equation (1). This measure is defined as6

( )t

c

t

wB t

itit

it ∂∂

−−∂

∂=

ln1

ln η (9)

where ηit is the expenditure elasticity of demand and Bit is the scale adjusted bias for the ith good

in quarter t. The biases were calculated at the estimated shares for each quarter, but because the

parameter u in equation (1) is unobservable it was necessary to use actual expenditures in

forming bias estimates.

6The equation presented by Antle and Capalbo is slightly different. They write:

t

C

Q

C

Q

w

t

wB ii

i ∂∂

∂

∂∂

∂−∂

∂=−

lnlnlnln1

However, replacing Q with u, differentiating equations (1) and (5), and substituting into the above formula yieldsequation (9).

12

The estimated structural change biases are shown in Figures 3 and 4. The appropriate

interpretation is of importance. Positive values imply that the structural change favors that

factor. A glance at the figures shows that the biases fluctuate greatly over the sample. However

fish biases have tended to be positive while beef, pork, and chicken have tended to be negative

throughout the sample. The biases for fish and pork have been relatively small over the sample

compared to beef and fish biases. The mean biases before 1976 are basically unchanged versus

those for the post 1976 period. The means for the entire sample are -0.0004, -0.0021, -0.0005,

0.0001 for beef, pork, chicken, and fish respectively. While these are small, the graph shows that

for individual sample periods some of the biases are large7. Moschini and Meilke (1989) found

somewhat different aggregate biases. In the cases of beef only the mean GFLS/AIDS bias has a

compatible sign with the Moschini and Meilke (1989) estimates. Only the pork bias estimates

are similar in magnitude. These differences may arise from the scale effect adjustments in the

GFLS/AIDS bias estimates or from differences in estimation techniques. The GFLS/AIDS

results suggest that, with the exception of fish, the periods of negative bias have outweighed the

positive.

The estimated elasticities derived from the GFLS/AIDS model are of also of interest in

evaluating how meat demand has evolved over time. The uncompensated elasticities were

calculated for each quarter (at the mean of the predicted shares) with the following formula

suggested by Green and Alston,

[ ] [ ] IIAIBCE −++= −1 (10)

where the typical elements are:

−

−=

i

ji

i

ijijij w

wwa βγδ in A;

=

i

ii wb β in B;

jjj pwc ln= in C; εij in E; and δij is one if i=j and zero otherwise. This formula is slightly

7The means of the absolute values of the biases were 0.0086, 0.0156, 0.060, and 0.0030.

13

different from that used by Chalfant and others but it has been shown, in Monte Carlo

experiments, to outperform the more traditional alternatives (see Alston, Foster, and Green).

To conserve space, only the own price elasticities are presented in Figures 5 and 6.

Figure 5 displays the time path of the own price elasticities of beef and chicken demand8. Notice

that the own price elasticity of beef and chicken trends upward or becomes less responsive to

price changes, with chicken experiencing a large decrease in responsiveness around 1974-1975.

To the contrary, the pork demand own price elasticity increases in absolute value or becomes

more responsive to price changes, while the fish own price elasticity increases in the later portion

of the sample. In the case of beef, the absolute change over the sample period is not that large

but is enough to have meaningful economic consequences. These elasticity estimates differ from

those of Moschini and Meilke (1989). They found that beef demand was more responsive and

pork demand was less responsive after the mid-1970's structural change and that chicken demand

did not change much. However, the GFLS/AIDS estimates for fish own price elasticities are in

agreement with those of Moschini and Meilke (1989), that is both suggest that the own price

elasticity for fish has become more responsive. Consistency between the two sets of results is

also mixed for the uncompensated cross-price elasticities. The GFLS/AIDS elasticity estimates

with the same sign and direction of change as those estimated by Moschini and Meilke (1989)

were: εbp, εbc, εbf, εcb, εfb, εfp, and εfc. The GFLS/AIDS estimates for εbp and εfp were similar in

sign but less pronounced than Moschini and Meilke’s (1989) results.

Determinants of Structural Change

In the previous sections, changes in the own and cross-price demand elasticities gave

evidence of changing consumption patterns. In this section, an attempt is made to determine the

14

significant causes of these changes. Two schools of thought have been suggested: changes in

preferences and changes in relative prices. These hypotheses suggest two competing

specifications for explaining intertemporal variations in the price elasticities of demand for meat.

The elasticities used here are slightly different from those discussed in the previous section. In

order to test these hypotheses the following non-nested models were specified.

Relative Prices Model:

ijDc

ff

c

pp

c

bbij uD

p

p

p

p

p

p+++++= αααααε 0 (11)

Cholesterol Index Model

ijDCHij vDCH +++= βββε 0 (12)

where the αi and βi are parameters to be estimated, CH is the Brown and Schrader cholesterol

index divided by its mean, D is a matrix of quarterly dummy variables for the first three quarters,

and the uij and vij are contemporaneously correlated disturbance terms. Note that since all

equations in each set (Relative Prices Model and Cholesterol Index Model) contains the same

regressors it is sufficient to estimate each of the sixteen equations of the form in (11) and (12) by

ordinary least squares9.

Significant autocorrelation was detected in each of the equations in (11) and (12). This

along with the quarterly dummy variables presents a difficulty with respect to a straightforward

application of the J-test. The original development by Davidson and McKinnon did not address

the possibility of autocorrelation. Anatonovitz and Green point out that the correction for

autocorrelation should be made during the estimation of the competing models, but that this

8All of the estimated own price elasticities (compensated and uncompensated) were negative for all quarters.Expenditure elasticities were also calculated. In addition, all of the estimated expenditure elasticities were positivefor all quarters.9 SUR becomes OLS when Xi = X for all equations.

15

information should not be included when calculating the predicted values used in the second step

of the J-Test procedure so as not to bias the test in favor of that model. The same can be said of

the quarterly dummy variables. Consequently, the equations in (11) and (12) were estimated by

the Prais-Winsten method to preserve the initial observation. The predicted values were

calculated by the general form:

tt xy βα ˆˆˆ += (13)

rather than

11ˆˆˆ −− +−+= tttt yxxy ρρβα (14)

The results of the J-Test for each of the sixteen uncompensated elasticities and the two

models are presented in Table 2. Notice that the cholesterol index model is rejected in 13 out of

16 cases. However, the model based on relative prices cannot be rejected in ten of the cases.

The only pattern for those that are not rejected seems to be from the elasticities taken with

respect to beef and pork, εbb, εbp, εbc, εbf, εpb, εpp, εpc, εpf, εcc, and εfc.

Conclusions

In this paper a demand system with time varying coefficients (GFLS/AIDS) was

developed and estimated using quarterly U.S. meat consumption data. Changes in structure were

embodied in the movement of parameter estimates over time. Even with an objective function

which penalized parameter movement fairly strong, the parameters achieved economically

significant changes over the sample period. Estimates of scale adjusted biases in structural

change exposed a fluctuating situation which has, on average, been biased against beef and in

favor of fish.

Of greater importance, however, was the discovery of non-random movements in the own

price elasticities of demand for beef, pork, and fish. In the case of pork, demand has become

16

more own price responsive while beef and chicken demand has become less responsive. These

trends have significant implications for meat industries. Less elastic beef demand suggests that

future cattle price cycles will be larger relative to their accompanying quantity cycles, assuming

no changes in the supply schedule. In fact, some of this may already be occurring and could help

explain the slow cattle herd build up of the early 90’s in the face of high prices. The beef

industry would appear to be in a good position to take advantage of an increasingly inelastic

demand for their product.

In order to determine the source(s) of changing consumption patterns over time, sets of

regressions were estimated using the uncompensated elasticities as dependent variables. The

independent variables were relative prices, the Brown and Schrader cholesterol index, and

quarterly dummy variables. The results strongly support a significant role for relative prices and

changing tastes and preferences. There seems to be less support for the relative price hypothesis

for chicken and fish demand elasticities and more for the beef and pork cases (see Table 2). In

other words, this lite beer, "tastes great and it's less filling!"

17

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19

Table 1. Summary of the GFLS Estimated Coefficients andAIDS/GFLS Estimated Own Price Elasticities

Coefficient Average Minimum Maximumαb 0.0273 0.0239 0.0290αp 0.0140 0.0117 0.0173αc 0.0069 0.0047 0.0107γbb 0.0163 0.0070 0.0267γbp 0.0134 0.0031 0.0198γbc 0.0159 0.0040 0.0218γbf -0.0457 -0.0533 -0.0273γpp 0.0066 -0.0007 0.0184γpc 0.0033 -0.0108 0.0131γpf -0.0234 -0.0413 -0.0082γpp 0.0187 0.0030 0.0342γpf -0.0379 -0.0449 -0.0321γff 0.1070 0.0769 0.1230βb 0.1109 0.0973 0.1166βp 0.0632 0.0582 0.0731βc 0.0224 0.0183 0.0310βf -0.1965 -0.2125 -0.1802εbb -0.4497 -0.4904 -0.4249εpp -1.3807 -1.4541 -1.3419εcc -0.9915 -1.1608 -0.8654εff -0.1407 -0.1511 -0.1320

a Subscripts b, p, c, and f denote beef, pork, chicken, and fish.

20

Table 2. J-Test Results for Relative Prices and Cholesterol Index Models of GFLS/AIDSUncompensated Demand Elasticities

Coefficients from the Relative PricesModel

Results of J-Test at 5% Level

Elasticity αb αp αf Relative PricesModel

CholesterolModel

εbbb 0.009*c

(0.003)d0.008*(0.006)

0.011(0.002)

can’t reject reject

εbp -0.003*(0.001)

-0.002(0.001)

-0.001*(0.001)

can’t reject reject

εbc -0.002(0.001)

-0.001(0.002)

-0.003*(0.001)

can’t reject reject

εbf -0.006*(0.002)

-0.006(0.003)

-0.004*(0.001)

can’t reject reject

εpb -0.022*(0.003)

-0.008(0.005)

0.009*(0.002)

can’t reject can’t reject

εpp 0.026*(0.009)

-0.082(0.015)

-0.011*(0.004)

can’t reject can’t reject

εpc -0.035*(0.007)

-0.008(0.011)

0.022*(0.003)

can’t reject reject

εpf -0.021*(0.003)

-0.004(0.006)

0.011*(0.002)

can’t reject reject

εcb -0.016*(0.002)

-0.009*(0.003)

0.007*(0.001)

reject reject

εcp -0.009*(0.002)

-0.011*(0.004)

0.010*(0.001)

reject reject

εcc 0.009(0.030)

0.181*(0.051)

0.036*(0.015)

can’t reject reject

εcf -0.027*(0.005)

-0.037*(0.008)

0.037*(0.002)

reject reject

εfb -0.005*(0.0001)

-0.003*(0.0001)

0.002*(0.0001)

reject can’t reject

εfp -0.003*(0.0001)

-0.002*(0.0001)

0.002*(0.0001)

reject reject

εfc -0.001*(0.0001)

-0.001(0.001)

0.001*(0.0003)

can’t reject can’t reject

εff -0.005*(0.001)

-0.006*(0.002)

0.003*(0.0006)

reject reject

b Subscripts b, p, c, and f denote beef, pork, chicken, and fish.c * indicates significance at the 5% level.d The numbers in parentheses are standard errors.

21

Figure 1. Estimated Own Price Coefficients from GFLS/AIDS Model for Beef and Chicken

0.006

0.012

0.018

0.024

0.030

1967 1970 1973 1976 1979 1982 1985 1988

Year-Quarter

Bee

f Ow

n P

rice

Coe

ffice

ints

0.00

0.01

0.02

0.03

0.04

Chi

cken

Ow

n P

rice

Coe

ffice

ints

Beef Chicken

Figure 2. Estimated Own Price Coefficients from GFLS/AIDS Model for Pork and Fish

-0.001

0.004

0.009

0.014

0.019

1967 1970 1973 1976 1979 1982 1985 1988

Year-Quarter

Por

k O

wn

Pric

e C

oeffi

cein

ts

0.07

0.08

0.09

0.10

0.11

0.12

0.13

Fis

h O

wn

Pric

e C

oeffi

cein

ts

Pork Fish

22

Figure 3. Structural Change Biases in Beef and Chicken Demand

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

1967 1970 1973 1976 1979 1982 1985 1988

Chicken Beef

Figure 4. Structural Change Biases in Pork and Fish Demand

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

1967 1970 1973 1976 1979 1982 1985 1988

Pork Fish

23

Figure 5. Beef and Chicken Own Price Elasticites

-1.180

-1.130

-1.080

-1.030

-0.980

-0.930

-0.880

1967 1970 1973 1976 1979 1982 1985 1988

Beef

Chi

cken

-0.50

-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

-0.42

Chicken

Bee

f

Figure 6. Pork and Fish Own Price Elasticites

-1.46

-1.44

-1.42

-1.40

-1.38

-1.36

-1.34

1967 1970 1973 1976 1979 1982 1985 1988

Pork

Por

k

-0.155

-0.150

-0.145

-0.140

-0.135

-0.130

-0.125

Fish

Fis

h